A biologist uses a microscope to view a sample of a plant cell. The microscope magnifies the image so that...
GMAT Geometry & Trigonometry : (Geo_Trig) Questions
A biologist uses a microscope to view a sample of a plant cell. The microscope magnifies the image so that any length on the image is \(50\) times the corresponding actual length of the cell. If the area of the plant cell's image as seen through the microscope is \(1{,}000\) square millimeters, what is the actual area of the plant cell, in square millimeters?
1. TRANSLATE the problem information
- Given information:
- Linear magnification = \(50\) (any length on image is \(50\) times actual length)
- Image area = \(1,000\) square millimeters
- Need to find actual area of the cell
2. INFER the relationship between linear and area scaling
- Key insight: When linear dimensions are magnified by factor k, area is magnified by factor \(\mathrm{k}^2\)
- Since linear magnification = \(50\), area magnification = \(50^2 = 2,500\)
- This means the image area is \(2,500\) times larger than the actual area
3. INFER the correct operation direction
- We have the magnified area \((1,000)\) and need the actual area
- Since magnification multiplies by \(2,500\), we must divide to reverse this:
- \(\mathrm{Actual\ Area} = \mathrm{Image\ Area} \div \mathrm{Area\ Scale\ Factor}\)
4. SIMPLIFY to find the answer
- \(\mathrm{Actual\ Area} = 1,000 \div 2,500\)
- \(\mathrm{Actual\ Area} = 10 \div 25 = 2 \div 5 = 0.4\) square millimeters
Answer: A. 0.4
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak INFER skill: Not recognizing that area scales by \(\mathrm{k}^2\), instead using the linear scale factor directly
Students often think: "The magnification is 50, so I divide 1,000 by 50 to get the actual area."
This reasoning misses that area (2-dimensional) scales differently than length (1-dimensional). Using \(50\) instead of \(2,500\) gives: \(1,000 \div 50 = 20\)
This may lead them to select Choice D (20)
Second Most Common Error:
Conceptual confusion about magnification direction: Thinking that actual area should be larger than image area
Some students get confused about which is bigger - the actual cell or the image. They might multiply instead of divide: \(1,000 \times\) something, leading to answers far outside the given choices.
This leads to confusion and guessing among the smaller answer choices.
The Bottom Line:
This problem tests whether students understand that magnification affects different dimensions (length vs. area) by different scale factors. The key insight is recognizing the \(\mathrm{k}^2\) relationship for area scaling.